Method for controlling an index modulation of a switchable polymer dispersed liquid crystal optical component

ABSTRACT

Described herein are the materials, mechanisms and procedures for optimizing various performance parameters of HPDLC optical devices in order to meet differing performance requirements. These optimization tailoring techniques include control and independent optimization of switchable HPDLC optical devices to meet the demanding requirements of anticipated applications for, inter alia, the telecommunications and display industries. These techniques include optimization of diffraction efficiency, i.e., index modulation, polarization dependence control, haze, cosmetic quality, control of response and relaxation time, voltage driving for on and off switching, and material uniformity. This control and independent optimization tailors properties of switchable HPDLC optical devices according to the specific requirements of the application of the switchable HPDLC optical device. The invention disclosed herein retains the desirable attributes of the multi-functional acrylate system for forming HPDLC optical devices, but adds new materials to the acrylate system and/or new process control to the recording to optimize performance parameters as may be needed for specific applications. This results in high optical quality switchable holograms with good diffraction efficiency and low, stable switching voltage.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a divisional of U.S. patent application Ser. No.11/129,441, filed May 16, 2005 now U.S. Pat. No. 7,072,020, which is adivisional of U.S. patent application Ser. No. 10/408,259, filed Apr. 8,2003, now U.S. Pat. No. 6,950,173. Each of the above-identifiedapplications is incorporated herein by reference in its entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The purpose of this invention is to control and optimize the performanceparameters of switchable holograms to tailor the properties toapplication-specific devices.

2. Description of the Related Art

U.S. Pat. No. 5,942,157 provides a description of materials and methodsfor producing switchable holographic Bragg gratings.

U.S. Pat. Nos. 5,751,452 and 5,748,272 to Tanaka et al. teach an opticaldevice made from a switchable holographic polymer dispersed liquidcrystal (hereafter “PDLC”) grating and methods for fabricating the same.Tanaka et al. teach the use of NOA65 (polyene and polythiol mixture),but do not teach how it may be used in conjunction with amultifunctional acrylate to reduce switching voltage and eliminatevoltage creep. In their teaching, NOA65 is the sole polymerizablemonomer in the described embodiments. These embodiments are also foundin U.S. Pat. Nos. 4,938,568 and 5,096,282 to Margerum et al.

U.S. Pat. No. 5,875,012, and European Patent Application Nos.98300541.1, 98300543.0, and 98300468.0 to Crawford et al. teachreflective displays made with switchable PDLC holograms, but providelittle in the way of materials or methods for optimizing performance.Crawford et al. teach the use of an anisotropic polymer index-matched tothe liquid crystal to reduce haze at large viewing angles. This is alsotaught in U.S. Pat. Nos. 4,994,204 and 5,240,636 Doane et al. U.S. Pat.No. 5,731,853 to Taketomi et al. and U.S. Pat. No. 6,083,575 Ninomiya etal. teach devices made with switchable PDLC holograms, but provide noteaching for optimizing switchable hologram performance.

U.S. Pat. No. 5,313,317 to Saburi et al. and U.S. Pat. Nos. 5,330,264and 5,648,857 to Ando et al. teach beam control methods for controllingunwanted gratings (i.e., “ghost holograms”) in non-PDLC holograms usingparticular geometrical arrangements.

Each of the above-identified references is incorporated by referenceherein in its entirety.

SUMMARY OF THE INVENTION Summary of the Problem

Demand for information has become a strong driver in many business,consumer, and government applications. Three key components of thisdemand are the storage, transmission, and display of information. Thelatter two in particular are placing severe demands on availablehardware and software. In communications, there has been an explosion oftraffic driven by the Internet, business data, and digital imagetransfers. In the end-point use of this huge data stream, visualutilization and management of data have high priority. Large datacontent requires high resolution (SVGA to XGA) along with full-colorcapability. The technological response to these challenges has spawnedseveral innovations. For telecommunications applications, part of thetechnological response is to provide higher data rates and bandwidthextension through the use of dense wavelength division multiplexing(DWDM). For easy visual access to information, portable and handhelddevices are evolving along with flat screens and personal displays. Inaddition, efforts are underway to make the advantages of DVD and HDTVavailable in these formats.

Optics is at the core of all of these technologies. The informationrevolution is placing stringent demands on several optical components.For example, short and long-period fiber Bragg gratings are playing keyroles in telecommunications, but the demand for multiple wavelengths andthe ability for dynamic reconfiguration by DWDM is growing. Ininformation display applications, the use of portable andmicro-displays, combined with virtual display technology, is creatingthe need for complex off-axis optical systems in very compact,lightweight packages. This becomes impossibly heavy and cumbersome withconventional refractive and reflective optics.

Diffractive optics is the natural response to many of these demands. Butthese devices are by their very nature monochromatic. Multi-wavelengthand dynamic reconfiguration capabilities are forcing a reconsiderationof the use and fabrication of diffractive optical elements to satisfythe growing needs of the information revolution.

Switchable holographic optical elements (HOEs) have been invented tofulfill the promise of diffractive optics in meeting the technologicalchallenges in telecommunications and information display. Multi-layeredswitchable holographic optical elements in a single solid-state deviceform a substitute for multiple static elements and complexrefractive/reflective optical systems. This dramatic innovation hasprompted one technology developer to coin the phrase “an optical systemin a chip” as an apt description of switchable HOEs.

To be successful, switchable hologram technology must present a flexibleapproach to optical element design and fabrication, offering highefficiency and optical quality with low power consumption. Moreover, itmust be tailored to customer specifications, i.e., it has to be veryapplication-specific. For example, devices in telecommunicationsapplications that require specific wavelength and format considerationsinclude reconfigurable add/drop switches, multiplexers, optical crossconnects, optical switches, wavelength selectors and tuners, andspectral attenuators or gain flatteners. Examples of such needs alsoabound in the information display area, including personal DVD/HDTVviewers, portable displays, data phone/handheld Internet displays,wearable PC displays, digital picture frames, desktop telephoneE-mail/Internet displays, ultra-portable projection systems, and desktopmonitors.

Summary of the Solution

Polymer-dispersed liquid crystal (“PDLC”) holographic materials have nowbeen successfully demonstrated in several components and prototypedevices. These components and devices offer a solution to the need foran electronically driven, multi-layer, multi-wavelength, complex opticalsystem in a thin, lightweight, low-electrical-power element. Thefabrication of switchable holograms by the photopolymerization-inducedphase separation of liquid crystal (“LC”) from an initially homogenouspre-polymer mixture has been discussed in commonly owned U.S. Pat. No.5,942,157. Prior to forming the hologram thereon, the pre-polymermixture consists of a multi-functional acrylate monomer (or mixture ofmulti-functional monomers of differing functionality) combined with amono-functional aromatic vinyl monomer and a LC, along with other keyingredients, including a photoinitiator dye. Similarly, the holographicrecording process has also been described, employing a single-stepmethod wherein coherent laser beams combine to form an interferogram inthe plane of the pre-polymer mixture. As the system cures, the LC phaseseparates to form the hologram, consisting of a pure grating or mixtureof gratings. These gratings are comprised of alternating LC-rich andpolymer-rich regions.

As these switchable holographic materials and devices near applicationin the markets discussed above, it is becoming clear that severalperformance parameters are critical to the success of the devicesemploying this technology. For example, various applications of theswitchable holographic-PDLC (hereafter “HPDLC”) optical elements requirepolarized light, while others require diffraction of unpolarized light.Consequently, there is an advantage to having the capability to controlthe polarization dependence of the PDLC grating for specificapplications.

Further, in many applications with holograms, haze is a problem. InHPDLC optical elements, haze is produced by light scattering frominhomogeneities in the HPDLC film component of the optical element. Someof these inhomogeneities are contaminants that can be controlled bycareful processing. Others, however, originate from the phase-separatedLC droplets. The diffraction planes themselves will produce some randomscattering due to nonuniform distributions of LC droplets from plane toplane. However, a major source of scattering comes from phase-separateddroplets that occur outside the desired Bragg planes. Examples of thisare cross-gratings and diffraction rings formed by spurious reflectionsand diffraction of the recording beams. Also, in some cases LC mayrandomly phase separate in the polymer-rich regions. Scattering is astrong function of droplet size and density. In some cases, a haze aslarge as 10% has been measured. It is strongly desired to reduce andcontrol the amount of haze in holograms for specific applications.

Further still, in electrically switchable holograms, minimization andcontrol of power dissipation is an important consideration. Powerdissipation leads to joule heating, which in some cases can causeproblems with thermal stability. Also, large power consumption requiresa more expensive electrical power supply and possibly larger voltages,which may lead to electrical shorting that destroys the hologram'susefulness. This depends largely on the switching voltage of thehologram. High switching voltage leads to large current drawn from thepower supply. In switchable PDLC gratings, power consumption anddissipation comes from current drawn to charge up the transparentelectrodes, as well as from resistive heating in the transparentelectrodes and through the hologram, due to a finite conductivity of thePDLC material.

Switching speed requirements of the HPDLC optical elements depends onthe intended application. Some applications may require on/off-switchingtimes in the microsecond regime, while some may only require millisecondresponse. Consequently, it is useful to have the ability to tailor theswitching speed to the application in order to optimize otherparameters, such as switching voltage.

Some applications for the HPDLC optical elements place the elements inharsh environments that degrade its properties. Typical environmentalparameters that prove deleterious to operation include temperature,humidity, and UV exposure, the most severe of these being temperature.LCs nominally have freezing points below 0° C. and nematic-to-isotropic(N-I) transition points at 65-100° C. The high temperature range isusually the most problematic in devices. Any contaminants or diluents inthe LC will lower the LC's order parameter and thereby reduce its N−1transition. This in turn can significantly reduce diffractionefficiency. For example, the N−1 transition may be reduced by as much as30-40° C. by such contaminants/diluents. This severely restricts theoperating temperature of the hologram. Consequently, the ability tocontrol the environmental vulnerability of the HPDLC optical elements isdesirable.

The current invention sets forth materials, mechanisms and proceduresfor optimizing various performance parameters in order to meet differingperformance requirements. These optimization tailoring techniquesinclude control and independent optimization of switchable HPDLC opticaldevices to meet the demanding requirements of anticipated applicationsfor, inter alia, the telecommunications and display industries. Thesetechniques include optimization of diffraction efficiency, i.e., indexmodulation, polarization dependence control, haze, cosmetic quality,control of response and relaxation time, voltage driving for on and offswitching, and material uniformity. This control and independentoptimization tailors properties of switchable HPDLC optical devicesaccording to the specific requirements of the application of theswitchable HPDLC optical device. The invention disclosed herein retainsthe desirable attributes of the multi-functional acrylate system forforming HPDLC optical devices, but adds new materials to the acrylatesystem and/or new process control to the recording to optimizeperformance parameters as may be needed for specific applications. Thisresults in high optical quality switchable holograms with gooddiffraction efficiency and low, stable switching voltage.

A first embodiment of the present invention describes a system forcontrolling the index modulation of a polymer dispersed liquid crystaloptical element. The system comprises a first substrate and a secondsubstrate with a pre-polymer liquid crystal material therebetween; and afirst and a second electrode pattern on each of the first and secondsubstrates, wherein at least one of the first and second electrodepatterns consists of interdigitated electrodes.

A second embodiment of the present invention describes a method forcontrolling the index modulation of a switchable polymer dispersedliquid crystal optical component. The method comprises providing apre-polymer liquid crystal material between a first and secondsubstrate, the first and second substrate having a first and secondelectrode pattern thereon, respectively, for applying a switchingvoltage to the switchable polymer dispersed liquid crystal opticalcomponent, wherein at least one of the first and second electrodepatterns consists of interdigitated electrodes; applying a voltageapproximately equal to the switching voltage to every otherinterdigitated electrode, creating an in-plane electric field within thepre-polymer liquid crystal material; holographically irradiating thepre-polymer liquid crystal material resulting in polymerization of thepre-polymer liquid crystal, wherein liquid crystal droplets formed fromthe holographic irradiation are formed with symmetry axes oriented inthe same direction as the in-plane electric field; and removing thevoltage approximately equal to the switching voltage once polymerizationis complete.

A third embodiment of the present invention describes an inverse modeswitchable grating system. The system comprises a holographicallypolymerized polymer dispersed liquid crystal material having aswitchable grating formed therein; and at least a first and a secondelectrode for applying a switching field to the switchable grating inorder to vary a diffraction efficiency thereof, wherein application ofthe switching field increases the diffraction efficiency of theswitchable grating and removal of the switching field decreases thediffraction efficiency of the switchable grating.

A fourth embodiment of the present invention describes a method forswitching a holographic diffraction grating via a switching fieldbetween a first diffraction efficiency and a second diffractionefficiency. The method comprises orienting the holographic diffractiongrating such that an internal angle of p-polarized light incidentthereon satisfies the following condition for a switching field of zero,

${\tan\;\theta_{\rho}} = ( \frac{ɛ_{xx}^{(1)}}{ɛ_{zz}^{(1)}} )^{1/2}$wherein θ_(ρ) is the angle of incidence of a reference wave of theincident light and ∈_(xx) ⁽¹⁾ and ∈_(zz) ⁽¹⁾ are the x and z componentsof the modulation of the dielectric tensor for a material comprising theholographic diffraction grating and the holographic diffraction gratinghas a first diffraction efficiency; and applying a switching fieldgreater than zero in order to switch the holographic diffraction gratingto a second diffraction efficiency.

A fifth embodiment of the present invention describes a method forsplitting a light beam. The method comprises receiving a light beam at aholographically polymerized polymer dispersed liquid crystal materialhaving an electrically controllable switchable grating formed therein;and controlling the application of an electric field to the switchablegrating, wherein when no electric field is applied to the switchablegrating the light beam is split into s-polarized light that is reflectedfrom the switchable grating and p-polarized light that is transmittedthrough the switchable grating and further wherein when a thresholdswitching electric field is applied to the switchable grating the lightbeam is split into s-polarized light that is transmitted through theswitchable grating and p-polarized light that is reflected from theswitchable grating.

A sixth embodiment of the present invention describes a method forcontrolling the haze in a holographically polymerized polymer dispersedliquid crystal optical element. The method comprises forming a looselygelled network within a pre-polymerized polymer dispersed liquid crystalmaterial and holographically polymerizing the polymer dispersed liquidcrystal material, including the loosely gelled network, to form thepolymer dispersed liquid crystal optical element with decreased haze.

A seventh embodiment of the present invention describes a method forforming a holographically polymerized polymer dispersed liquid crystaloptical element with reduced haze. The method comprises adding apre-polymerized polymer dispersed liquid crystal material to apre-existing loosely gelled network; placing the pre-existing looselygelled network containing the pre-polymerized polymer dispersed liquidcrystal material between first and second transparent substrates; andinterfering a first beam and a second beam within the pre-existingloosely gelled network containing the pre-polymerized polymer dispersedliquid crystal material to form the holographically polymerized polymerdispersed liquid crystal optical element with reduced haze.

An eighth embodiment of the present invention describes a method fordriving a polymer dispersed liquid crystal hologram. The methodcomprises providing a polymer dispersed liquid crystal hologram betweena first and second substrate, the first and second substrate having afirst and second electrode pattern thereon, respectively, for applying aswitching voltage to the polymer dispersed liquid crystal hologram,wherein the first and second electrode patterns consist ofinterdigitated electrodes; applying a first voltage scheme, wherein avoltage approximately equal to the switching voltage is applied to theinterdigitated electrodes on the first substrate and the interdigitatedelectrodes on the second substrate are connected to ground in order todrive the polymer dispersed liquid crystal hologram off; and removingthe first voltage scheme and applying a second voltage scheme, wherein avoltage approximately equal to the switching voltage is applied to everyother interdigitated electrode on the first and second substrates andthe intermittent electrodes therebetween are connected to ground inorder to drive the polymer dispersed liquid crystal hologram on.

BRIEF DESCRIPTION OF THE FIGURES

In the Figures:

FIG. 1 shows a conventional diffraction geometry for a Braggtransmission grating;

FIG. 2 shows a model depicting the diffraction and polarizationproperties of light in a HPDLC optical device according to an embodimentof the present invention;

FIG. 3 shows the distribution of symmetry axes in LC domains of a HPDLCoptical element according to an embodiment of the present invention;

FIG. 4 shows the orientation of a LC domain symmetry axis in thepresence of an electric field according to an embodiment of the presentinvention;

FIG. 5 shows the orientation of an LC droplet director N relative to alaboratory reference frame xyz according to an embodiment of the presentinvention;

FIGS. 6 a and 6 b show the form of distribution functions used in ananalysis of certain embodiments of the present invention;

FIGS. 7( a) through 7(c) show the LC droplet direction distributionsaccording to an embodiment of the present invention;

FIGS. 8( a) and 8(b) show coupled wave interaction for reflection andtransmission gratings, respectively, according to embodiment of thepresent invention;

FIG. 9 shows electric field effects on diffraction efficiency of agrating at normal incidence according to an embodiment of the presentinvention;

FIG. 10 shows a switching curve of a grating at normal incidenceaccording to an embodiment of the present invention;

FIG. 11 shows the difference in switching behavior between normal andoff-incidence radiation on a grating according to an embodiment of thepresent invention;

FIG. 12 shows the switching of p-polarized and s-polarized lightaccording to an embodiment of the present invention;

FIGS. 13 a and 13 b show electrode configurations for aligning LCdroplets during formation of a HPDLC optical element according to anembodiment of the present invention;

FIG. 14 shows a voltage scheme for aligning the LC droplets of a HPDLCoptical element according to an embodiment of the present invention;

FIG. 15 shows a conventional transmission hologram prepared withoutpre-exposure;

FIG. 16 shows a diffraction efficiency comparison according to anembodiment of the present invention;

FIG. 17 shows a hologram recording set-up according to an embodiment ofthe present invention;

FIGS. 18 a and 18 b show voltage schemes for switching a HPDLC opticalelement according to an embodiment of the present invention; and

FIG. 19 shows a voltage drive waveform for switching a HPDLC opticalelement according to an embodiment of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS OF THE PRESENTINVENTION

The preferred embodiments of the present invention utilize the materialsand/or process controls as set forth in the second column of Table 1 inorder to optimize the corresponding performance parameters of the firstcolumn of Table 1.

TABLE 1 Performance Parameters Materials/Process Control Diffractionefficiency Nematic director control (index modulation) Fringestability/contrast Gel network pre-stabilization Polarization dependenceNematic director control Haze Gel network pre-stabilization Indexmatching (Fresnel reflections & scattering) Cosmetic quality Gel networkpre-stabilization Index matching (Fresnel reflections & scattering)Inverse mode switching Anisotropic grating parameters Switching contrastratio LC droplet size/shape (dynamic range) Response/relaxation time LCdroplet size/shape Electrode design/voltage drive scheme

In order to provide a context for the implementation of the optimizationmaterials and techniques of the preferred embodiments of the presentinvention, the base features of the HPDLC optical devices are describedbelow. The HPDLC optical devices consist of a homogeneous mixture ofingredients (i.e., “pre-polymer material”) that includes the following:a polymerizable monomer (mixture of multi-functional acrylates,including at least a pentaacrylate), liquid crystal (“LC”) material(typically a mixture of cyanobiphenyls), a photoinitiator dye (one dyewith absorption spectrum overlapping recording laser wavelength), aco-initiator, a reactive diluent (formerly called cross-linking agent),and a surfactant-like additive (formerly called surfactant). Specificexamples of the homogeneous mixtures, as well as other formation processand material descriptions supporting the embodiments described hereinare found in U.S. Pat. No. 5,942,157 and U.S. patent application Ser.Nos. 09/033,512 entitled “Switchable Volume Hologram Materials andDevices,” filed Mar. 2, 1998; 09/033,513 entitled “Switchable VolumeHologram Materials and Devices,” filed Mar. 2, 1998; 09/033,514 entitled“Switchable Volume Hologram Materials and Devices,” filed Mar. 2, 1998;09/034,014 entitled “Switchable Volume Hologram Materials and Devices,”filed Mar. 2, 1998; 09/429,645 entitled “Switchable Volume HologramMaterials and Devices,” filed Oct. 29, 1999; 09/347,624 entitled“Switchable Volume Hologram Materials and Devices,” filed Jul. 2, 1999;09/363,169 entitled “Electrically Switchable Optical Couplers andReconfigurable Optical Polymer Dispersed Liquid Crystal MaterialsIncluding Switchable Optical Couplers and Reconfigurable Optical,” filedJul. 29, 1999; 09/742,397 entitled “Switchable Polymer-Dispersed LiquidCrystal Optical Elements,” filed Dec. 22, 2000; 09/577,166 entitled“Volume Hologram Replication System and Method for Replicating VolumeHolograms,” filed May 24, 2000; and 10/303,927 entitled “TailoringMaterial Composition for Optimization of Application-Specific SwitchableHolograms” filed Nov. 26, 2002; and U.S. patent application Ser. Nos.09/033,512, 09/033,513, 09/033,514, 09/034,014, 09/429,645, 09/347,624,and 09/363,169, each of which is incorporated by reference herein in itsentirety.

When the pre-polymer material is irradiated holographically, thephotoinitiator absorbs light in the bright fringes and reacts with theco-initiator, creating free radicals. The free radicals then initiatepolymerization of the multi-functional acrylates. The free-radicalprocess is very fast, and a three-dimensional polymer network is createdin just a few seconds. This rapid development of a densely cross-linkednetwork is critical to the phase separation of small LC droplets in thedark fringes, which is what establishes the hologram. Highly functionalacrylates are needed in order to produce this with a minimal exposuretime. A short exposure time is important in holography to reduce theeffects of unwanted vibrations and other perturbations that tend to washout the index modulation, and to make the process more amenable to massproduction. In addition, the resulting rapid polymerization and phaseseparation are favorable for the formation of small LC droplets, whichreduces random scattering losses (i.e., haze). The surfactant furthercontributes to reducing LC droplet size, yielding an optically clearhologram.

The three-dimensional network that results from the acrylatescontributes to “squeezing” the LC out into a separate phase and toyielding desirable optical properties for the hologram. In fact, thehologram and its switchability would not be possible without theseelastic attributes of the polymer. However, these strong elastic forcesalso make the polymer matrix very stiff. The stiffness contributes to ahigh switching voltage for the HPDLC optical devices. Moreover, themulti-functionality leads to continual post-polymerization after thehologram recording is completed. This stiffens the matrix further andslowly drives the switching voltage up. This is referred to as voltagecreep. In some cases the voltage creep can increase the switchingvoltage by as much as 100%. The elastic relaxation of themulti-functional acrylate system also produces another phenomenon:shrinkage. This can be seen in reflection holograms where the wavelengthof the Bragg reflection peak will blue shift due to shrinkage of thegrating period. Although not completely understood, it appears thatshrinkage has a major effect in a direction parallel to the gratingvector. Hence, in reflection gratings, this reduces the grating period,which can in most cases be compensated by appropriate recordingconditions (e.g., compensating recording angles). However, there appearsto be a more deleterious effect in transmission holograms where thegrating vector is in the plane of the hologram. Shrinkage in the planeof the hologram while recording appears to produce non-uniformity in thehologram. The shrinkage is not uniform, but creates elastic instabilityin the system, causing it to deform. This can almost be described as a“wrinkling” of the hologram with concomitant non-uniformity in thediffraction efficiency and cosmetic defects in the hologram'sappearance.

HPDLC optical devices also exhibit unique polarization dependence. InFIG. 1, we show incident and diffracted beams with two differentpolarization states: (a) perpendicular to the plane containing theincident, diffracted, and grating wavevectors (commonly known ass-polarization), and (b) in this plane (commonly known asp-polarization). In prior art switchable Bragg transmission grating 10,an incident beam of light 12 is deflected by a diffraction grating 14over a considerable angle that is equal to twice the Bragg angle for thewavelength of incident light, producing a diffracted exit beam 16. It iswell known that for an ordinary grating, s-polarized light will have astronger coupling (and hence larger diffraction efficiency) thanp-polarized light. The reason is that there is a complete overlap of theelectric field vectors for the incident and diffracted waves fors-polarization independent of angle of incidence. The overlap ofp-polarized beams depends on the angle between the two beams, going fromcomplete overlap for 0° angle to zero overlap for a 90° angle. Hence,for an ordinary grating, the diffraction efficiency of p-polarized lightshould never exceed that of s-polarized light.

However, in PDLC gratings of the type in FIG. 1 the opposite occurs: thediffraction efficiency of p-polarized light always exceeds that ofs-polarized light. Therefore, in the type of PDLC grating considered inFIG. 1, there is a built-in anisotropy that favors diffraction of lightpolarized in the plane containing the wavevectors and the gratingvector, even though the overlap of field vectors is smaller for thiscase than for the perpendicular polarization.

A simple model can explain these results, the LC phase separates asuniaxial domains 20 with symmetry axis pointed preferentially along thegrating vector 22 as shown in FIG. 2. The resulting domain 20 has anextraordinary index of refraction n_(e) along this symmetry axis, and asmaller ordinary refractive index n_(o) perpendicular to the axis. Sincep-polarized light has a component of its electric field along thesymmetry axis, it sees a refractive index heavily weighted by n_(e), andthus sees a relatively large index modulation. On the other hand,s-polarized light sees a refractive index weighted more by n_(o), andhence experiences a relatively small index modulation (n_(e)>n_(o)).Experimentally, the diffraction efficiency of s-polarized light isconsiderably weaker than that of p-polarized light. The symmetry axes ofLC domains 20 are not perfectly aligned with the grating vector 22.There is some small statistical distribution 25 of the axes about thisdirection. The average of the statistical distribution 25 points alongthe grating vector 22 as shown in FIG. 3. The average points along thegrating vector. Thus, s-polarized light will see a small amount of n_(e)mixed in with n_(o), which is what gives it its weak but measurablediffraction efficiency. We note that when a strong electric field 24 isapplied perpendicular to the plane of the grating vector 22, as shown inFIG. 4, nearly all LCs reorient in a direction along the beampropagation for some field value, and both s and p-polarized light seethe same index in the LC domains 20, approximately equal to n_(o). Sincethis index nearly matches the index of the surrounding polymer, theindex modulation for both polarization states disappears. The grating issaid to be switched “off” Additionally, as the field strength is furtherincreased, the LCs will eventually orient parallel to the field and thusnot be in an orientation to yield zero index modulation. Hence, thediffraction efficiency goes through a minimum near zero and thenincreases slightly with increasing field.

LC droplets form as nanoscale domains in HPDLC gratings. Detailedstudies by scanning electron microscopy (SEM) have revealed that thesedomains can be roughly ellipsoidal, but are quite often irregularlyshaped. The nematic configuration of LC molecules in micrometer scaledroplets has been successfully predicted in computer simulations andobserved by optical microscopy. A common arrangement of nematicdirectors in a spherical droplet is the so-called bipolar configuration,which has an axis of symmetry along a diameter and two point defects atthe polls. Computer simulations reveal that a similar pattern isobtained in slightly elongated droplets. The nematic configuration innanoscale domains is more elusive. However, nuclear magnetic resonancespectroscopy of deuterated-LC samples suggests that LC domains maycontain a line defect along their long axes. Optically, these dropletsappear to possess an axis of symmetry and behave as uniaxial domains.

A model was developed for the behavior of elongated LC dropletssubjected to an electric field as described in Wu et al. Liq. Cryst. 5,1453 (1989) which is incorporated herein by reference in its entirety.This model ignored the details of the nematic configuration and assumedthat the droplet is characterized by a single vector, which they calledthe droplet director N. In the absence of an electric field, N coincideswith the symmetry axis, which is also the major axis of the elongateddroplet. This uniaxial domain has a dielectric anisotropy that isapproximately equal to that of the bulk LC. Therefore, in the presenceof an electric field N will attempt to reorient along the direction ofthe applied field. This reorientation is resisted by an elastic torque,which arises due to the elastic distortion produced by the appliedfield. The elastic torque is related to the local radius of curvatureand some average elastic force constant of the droplet. A newequilibrium orientation of N is established by the condition that theelastic restoring torque balances the electrical torque. It is assumedthat nanoscale LC droplets can be treated in the same way and this modelis also applied to HPDLC gratings.

Assuming the LC droplets are uniaxial domains, they can be characterizedby a diagonal dielectric tensor in the reference frame of the droplet.Let ∈_(⊥) and ∈_(∥) be the dielectric constants perpendicular andparallel to N, respectively. At optical frequencies, the respectiverefractive indices are given by n_(⊥)=(∈_(⊥)/∈₀)^(1/2) andn_(∥)=(∈_(∥)/∈₀)^(1/2), where ∈₀ is the permittivity of free space.Because the nematic director configuration is not uniform within thedroplet, it is expected that n_(⊥)>n_(o) and n_(∥)<n_(e), where n_(o)and n_(e) are the ordinary and extraordinary refractive indices,respectively, of the bulk LC.

The orientation of the droplet director N relative to a laboratoryreference frame xyz is illustrated in FIG. 5. The electric field isassumed to be applied along the z axis as shown. The orientation of N isdescribed by spherical angles θ and 4:1 in the laboratory frame. Themodel of Wu et al. predicts that the field dependent equilibrium angleof N is given by

$\begin{matrix}{{\theta( {u,E} )} = {\frac{1}{2}{\tan^{- 1}\lbrack \frac{2u\sqrt{1 - u^{2}}}{{2u^{2}} - 1 + ( {E/E_{c}} )^{2}} \rbrack}}} & (1)\end{matrix}$where u=cos θ₀, with θ₀ the polar angle in the absence of an appliedfield, E is the electric field strength, and E_(c) is a critical fieldfor switching. Notice that the azimuth angle φ is unchanged by thefield.

In the laboratory reference frame the droplet dielectric tensor ∈_(d) isgiven by

$\begin{matrix}{ɛ_{d} = {{{\mathbb{R}}^{- 1}\begin{pmatrix}ɛ_{\bot} & 0 & 0 \\0 & ɛ_{\bot} & 0 \\0 & 0 & ɛ_{}\end{pmatrix}}{\mathbb{R}}}} & (2)\end{matrix}$where R is the rotation matrix that transforms the laboratory coordinateframe into the droplet coordinate frame, given by

$\begin{matrix}{{\mathbb{R}} = \begin{pmatrix}{\cos\;\theta\;\cos\;\phi} & {\cos\;\theta\;\sin\;\phi} & {{- \sin}\;\theta} \\{{- \sin}\;\phi} & {\cos\;\phi} & 0 \\{\sin\;\theta\;\cos\;\phi} & {\cos\;\theta\;\sin\;\phi} & {\cos\;\theta}\end{pmatrix}} & (3)\end{matrix}$and R⁻¹=R^(T) is the inverse (transpose) of R. Explicitly, the dropletdielectric tensor is

$\begin{matrix}{ɛ_{d} = \begin{pmatrix}{ɛ_{\bot} + {{\Delta ɛ}\;\sin^{2}\theta\;\cos^{2}\phi}} & {{\Delta ɛ}\;\sin^{2}\theta\;\sin\;{\phi cos}\;\phi} & {{\Delta ɛ}\;\sin\;{\theta cos}\;\theta\;\cos\;\phi} \\{{\Delta ɛ}\;\sin^{2}\;{\theta sin\phi}\;\cos\;\phi} & {ɛ_{\bot} + {{\Delta ɛsin}^{2}\theta\;\sin^{2}\phi}} & {{\Delta ɛ}\;\sin\;{\theta cos}\;{\theta sin\phi}} \\{{\Delta ɛ}\;\sin\;{\theta cos\theta cos\phi}} & {\Delta ɛsin\theta cos\theta sin\phi} & {ɛ_{\bot} + {{\Delta ɛcos}^{2}\theta}}\end{pmatrix}} & (4)\end{matrix}$where Δ∈=∈_(∥)−∈_(⊥), θ is given by Eq. (1), and φ=φ₀ and φ₀ areconstants for a given droplet. The droplet directors N for an ensembleof droplets are distributed about some mean orientation direction givenby ū and φ ₀ relative to the laboratory reference frame. The effectivetensor modulation seen by light will be related to an average over thisensemble. The azimuth angle range is restricted from 0 to π since therange from 0 to 2π includes −N, which is equivalent to N. In thisanalysis it is assumed that N has a symmetric distribution about θ=π/2and φ=π/2. In an alternative embodiment, the orientational distributionmay be skewed if some external force (e.g., shear) is applied to orientthe droplets preferentially in some particular direction. In the presentembodiment, it is assumed that the distribution function formsnaturally, with no external influence, and has a symmetry that isdictated by the direction of the grating vector and the naturallyoccurring orientation of droplet directors relative to this vector.Additionally, in cases of slanted or curved gratings it may not bepossible to assume symmetric orientational distributions, consequently,for these gratings additional assumptions must be made. The modeldescribed herein is useful to describe unslanted, planar reflection andtransmission gratings. The immediate consequence of a symmetricdistribution function is that all off-diagonal elements of the averagedroplet dielectric tensor vanish. This is because the off-diagonalelements in Eq. (4) are odd in either θ or φ about π/2.

For any given field strength E, each droplet will independently assume anew equilibrium orientation, described by Eq. (1), that is parameterizedby its initial polar angle θ₀. The azimuth angle will remain a constantdetermined by its initial value φ₀. Hence the average tensor elementscan be found by averaging over the initial orientation angles θ₀ and φ₀by factoring the distribution function into two functions, one dependenton θ₀ (or u) only and one dependent on φ₀ only. Calling these normalizeddistribution functions p(u) and q(φ₀), respectively, the averagediagonal tensor elements for the droplet can be written as

$\begin{matrix}{\langle ɛ_{dx} \rangle = {ɛ_{\bot} + {{\Delta ɛ}{\int_{0}^{\pi}{\int_{- 1}^{1}{\sin^{2}{\theta( {u,E} )}\cos^{2}\phi_{0}{p(u)}{q( \phi_{0} )}{\mathbb{d}u}\ {\mathbb{d}\phi_{0}}}}}}}} & (5) \\{\langle ɛ_{dy} \rangle = {ɛ_{\bot} + {{\Delta ɛ}{\int_{0}^{\pi}{\int_{- 1}^{1}{\sin^{2}{\theta( {u,E} )}\sin^{2}\phi_{0}{p(u)}{q( \phi_{0} )}\ {\mathbb{d}u}\ {\mathbb{d}\phi_{0}}}}}}}} & (6) \\{\langle ɛ_{dz} \rangle = {ɛ_{\bot} + {{\Delta ɛ}{\int_{- 1}^{1}{\cos^{2}{\theta( {u,E} )}{p(u)}\ {\mathbb{d}(u)}}}}}} & (7)\end{matrix}$It can be seen that these tensor elements depend on E through θ(u,E).

The form of the distribution functions is not known a priori and must beguessed. However, it is generally true that when dealing with a largenumber of statistically independent objects, the statistics of theensemble tend to approximately obey a Gaussian distribution.Accordingly, make the assumption that

$\begin{matrix}{{p(u)} = {A\;{\exp( {- \frac{( {u - \overset{\_}{u}} )^{2}}{2\sigma_{u}^{2}}} )}}} & (8) \\{{q( \phi_{0} )} = {B\;{\exp( {- \frac{( {\phi_{0} - {\overset{\_}{\phi}}_{0}} )^{2}}{2\sigma_{\phi}^{2}}} )}}} & (9)\end{matrix}$where ū ( φ ₀) is the mean value of u (φ₀), σ_(u) (σ_(φ)) is thestandard deviation of the u (φ₀) distribution, and A and B areappropriate normalization constants. Since the variables in this caseare periodic and hence do not extend to ±∞, care must be taken indefining the normalization constants. If the standard deviation issmall, the limits of integration in Eqs. (5)-(7) may be extended to ±∞though, and the distributions will look like ordinary Gaussianfunctions. However, to retain the possibility that the standarddeviations are not that small and that the distributions may tend towardconstant values representing isotropic orientation functions, computethe normalization constants by integrating Eqs. (8) and (9) over theappropriate range of variables and set the values equal to 1. FIGS. 6 aand 6 b illustrate the form of the distribution functions used in thisanalysis. For example, if the distributions are centered about meanvalues ū=0 and φ ₀=π/2, then the normalization constants in Eqs. (8) and(9) become

$\begin{matrix}{\begin{matrix}{A = \frac{2}{\sqrt{2\pi}{\sigma_{u}\lbrack {{{erf}( \frac{1 + \overset{\_}{u}}{\sqrt{2}\sigma_{u}} )} + {{erf}( \frac{1 - \overset{\_}{u}}{\sqrt{2}\sigma_{u}} )}} \rbrack}}} \\{B = {\frac{2}{\sqrt{2\pi}{\sigma_{\phi}\lbrack {{{erf}( \frac{{\overset{\_}{\phi}}_{0}}{\sqrt{2}\sigma_{\phi}} )} + {{erf}( \frac{\pi - {\overset{\_}{\phi}}_{0}}{\sqrt{2}\sigma_{\phi}} )}} \rbrack}}{\mspace{14mu}\;}{where}}}\end{matrix}\mspace{11mu}} & (10) \\{{{erf}(s)} = {\frac{2}{\sqrt{\pi}}{\int_{0}^{s}{{\exp( {- t^{2}} )}{\mathbb{d}t}}}}} & (11)\end{matrix}$is the error function. Taking this approach, the values of the means andstandard deviations can be varied independently to study the effects ondiffraction efficiency and switching. They also give an intuitiveinterpretation of the droplet director distribution that is easy tovisualize. For example, three distributions are shown in FIGS. 7( a),7(b) and 7(c), for an isotropic distribution shown in FIG. 7( a),orientations clustered about the x axis, i.e., small σ_(u) and σ_(φ), asshown in FIG. 7( b), and about the xy plane, i.e., small σ_(u) butisotropic in φ₀, as shown in FIG. 7( c).

For a mixture of two homogeneous, isotropic materials the effectivedielectric constant of the medium can be expressed approximately by∈=∈_(a)+f(∈_(b)−∈_(a)), where ∈_(a) and ∈_(b) are the dielectricconstants of the host and dispersed materials, respectively, and f isthe volume fraction of the dispersed material. This approximation holdsas long as ∈_(a)≈∈_(b). It is assumed that this relation can be appliedto an anisotropic HPDLC medium as long as the dielectric tensors of thetwo materials are nearly equal component by component. For the HPDLCgrating, the volume fraction f has the form of a periodic rectangularwave that is zero in the solid polymer region, and has a value f_(c) inthe PDLC region. The width of the PDLC region is αΛ, where α is afraction (0≦α≦1) and Λ is the grating period. To apply coupled wavetheory this distribution is Fourier analyzed, keeping terms up to firstorder. The spatially periodic dielectric tensor can thus be written as

$\begin{matrix}{{ɛ({??})} = {ɛ^{(0)} + {ɛ^{(1)}{\cos( {{??} \cdot {??}} )}\mspace{14mu}{where}}}} & (12) \\{ɛ^{(0)} = {{( {1 - {\alpha\; f_{c}}} )ɛ_{p}} + {\alpha\; f_{c}\langle ɛ_{d} \rangle}}} & (13) \\{ɛ^{(1)} = {\frac{2f_{c}}{\pi}{\sin({\alpha\pi})}( {\langle ɛ_{d} \rangle - ɛ_{p}} )}} & (14)\end{matrix}$In these equations, K is the grating vector (|K|=2π/Λ), <∈_(d)> is theaverage LC droplet dielectric tensor, and ∈_(p) is the polymerdielectric tensor. Assume that the polymer is isotropic so(∈_(p))_(ij)=∈_(p)δ_(ij), where ∈_(p) is a scalar. Since <∈_(d)> isdiagonal, ∈⁽⁰⁾ and ∈⁽¹⁾ are also diagonal. Hence the laboratory framealso serves as the principal axes frame of the medium. Equation (13)yields the principal refractive indices of the medium, n_(i)=(∈⁽⁰⁾_(ii)/∈₀)^(1/2) (i=x,y,z). In general, the medium is biaxial(n_(x)≠n_(y)≠n_(z)) and electro-optical through the dependence of<E_(d)> on E. The components of Eq. (14) are the dielectric tensormodulation elements that couple polarized optical waves in thediffraction grating.

The coupled wave theory of Kogelnik et al., Bell Syst. Tech. J. 48, 2909(1969) was recently extended to anisotropic media G. Montemezzani etal., Phys Rev. E. 55, 1035, (1997). Both of the references areincorporated herein by reference. The interaction of coupled waves inthick reflection and transmission holograms is illustrated in FIGS. 8(a) and 8(b). For a reflection grating, the grating vector is along the zaxis as shown in FIG. 8( a), while for a transmission grating it isalong the x axis as shown in FIG. 8( b). A field is applied along the zdirection in both cases. In the Bragg regime, only the signal wave (σ)and the reference wave (ρ) couple substantially. For an s-polarizedwave, i.e., polarization perpendicular to the plane of incidence, theoptical field vector is along they axis. For p-polarized light, i.e.,polarization in the plane of incidence, the field vector lies in the xzplane. The angle of incidence of the reference wave is θ_(ρ) while theangle of diffraction of the signal wave is θ_(σ). These angles refer tothe directions of the Poynting vectors of the reference and signalwaves, respectively. A diagonal modulation tensor cannot couples-polarized and p-polarized waves. Hence the signal and reference waveswill have the same type of polarization (i.e., s or p). The couplingcoefficient may be written as

$\begin{matrix}{\kappa = \frac{\pi\;{{\hat{e}}_{\sigma} \cdot ɛ^{(1)} \cdot {\hat{e}}_{\rho}}}{2ɛ_{0}{ng}\;\lambda\sqrt{{{\cos\;\theta_{\sigma}}}\cos\;\theta_{\rho}}}} & (15)\end{matrix}$where ê_(σ) (ê_(ρ)) is the unit vector of polarization for the signal(reference) wave, andn=√{square root over (n _(σ) n _(ρ))}  (16)with n_(σ) (n_(ρ)) the refractive index for the signal (reference) wave.The parameter g is related to the walk-off angle δ between the Poyntingvector and wave vector and is given byg=√{square root over (cos δ_(σ) cos δ_(ρ))}  (17)For unslanted gratings, n_(σ)=n_(ρ) and cos δ_(σ)=cos δ_(ρ). Notice thatfor an unslanted reflection grating θ_(σ)=π−θ_(ρ), so cos θ_(σ)≦0. Foran unslanted transmission grating θ_(σ)=2π−θ_(ρ) and cos θ_(σ)≧0. Inunslanted gratings |cos θ_(σ)|=cos θ_(ρ). The explicit expression of thecoupling coefficient for s polarization is

$\begin{matrix}{\kappa_{s} = \frac{{\pi ɛ}_{yy}^{(1)}}{2ɛ_{0}n_{y}g_{s}{\lambda cos\theta}_{\rho}}} & (18)\end{matrix}$with g_(s)=1, while for p polarization

$\begin{matrix}{\kappa_{p} = {\frac{\pi( {{ɛ_{xx}^{(1)}\sin^{2}\theta_{\rho}} - {ɛ_{zz}^{(1)}\cos^{2}\theta_{\rho}}} )}{2ɛ_{0}{n( \theta_{\rho} )}g_{p}{\lambda cos\theta}_{\rho}}\mspace{14mu}{with}}} & (19) \\{\lbrack {n( \theta_{\rho} )} \rbrack^{- 2} = {{n_{x}^{- 2}\cos^{2}\theta_{\rho}} + {n_{z}^{- 2}\sin^{2}{\theta\rho}\mspace{14mu}{and}}}} & (20) \\{g_{p} = \frac{{n_{x}^{2}\cos^{2}\theta_{\rho}} + {n_{z}^{2}\sin^{2}\theta_{\rho}}}{\lbrack {( {{n_{x}^{2}\cos^{2}{\theta\rho}} + {n_{z}^{2}\sin^{2}\theta_{\rho}}} )^{2} + {( {n_{x}^{2} - n_{z}^{2}} )^{2}\sin^{2}\theta_{\rho}\cos^{2}\theta_{\rho}}} \rbrack^{1/2}}} & (21)\end{matrix}$For weakly birefringent media (n_(x)≈n_(z)), g_(p)≈1.

In the case of reflection gratings, the peak diffraction efficiencyη_(j) (j=s or p) of a reflection grating isη_(j)=tan h ²(κ_(j) L)  (22)At off-normal incidence (θ_(ρ)≠0) the coupling coefficient is given byEq. (18) for s polarization and Eqs. (19)-(21) for p polarization.

Most of the LCs used in this analysis have similar values of n_(o) andn_(e). Hence the variation in diffraction efficiency between differentsystems is primarily due to the parameters α and f_(c), which arerelated to LC solubility for various types of gratings and polymersystems, and to the distribution functions p(u) and q(φ₀), which alsoappear to be dependent on the type of grating and LC. In order to limitthe variation of parameters, the index n_(p) can be measured for thepolymer and generally it is in the range of 1.52-1.54, depending on theamount of LC remaining in solution in the polymer. For s-polarizedlight, switching the grating to minimum diffraction efficiency impliesthat n_(⊥)≈n_(p) (see the discussion below). Hence the parameter n_(i)can be fixed by this condition. The quantities α and f_(c) can beestimated from SEM studies of HPDLC gratings. This leaves n_(∥) anddroplet statistics, i.e., means ū and φ ₀ and standard deviations σ_(u)and σ_(φ), as adjustable parameters to model experimental results.Information about statistical parameters can be obtained by observingthe polarization dependence of the grating. In a particular embodimentfor a reflection grating, the following values are selected forspecified parameters: n_(p)=1.530, n_(⊥)=1.535, n_(∥)=1.680, α=0.3,f_(c)=0.7. For these reflection gratings, a Bragg wavelength of 0.525 μmand a grating thickness L=8 μm are selected for this specific exemplaryembodiment. For these values, the effect of an electric field on thespectral diffraction efficiency of a Bragg grating with light at normalincidence is illustrated in FIG. 9. For this example, the mean dropletdirector is at π/2 radians with respect to the grating vector (i.e.,with respect to the z axis, ū=0) with an isotropic distribution ofdroplet directors in the xy plane (see FIG. 7 c). The standard deviationσ_(u) was selected to be 0.3. This implies that the droplet directorsexhibit a preferential ordering tangential to the grating plane. Thisseems be a tendency of the Merck TL series of LCs (e.g., TL213).Symmetry about the grating vector would seem to imply that there shouldbe no preferential direction of ordering in the plane of the grating.Therefore, at normal incidence the diffraction efficiency should beindependent of polarization. Switching occurs for an applied field˜E_(c). As the grating switches, the peak of the reflection notch shiftstoward the blue. This is due the field dependence of n_(y) (or n_(x)),which decreases with increasing field (see Eqs. (1), (5), (6), and(13)).

The width of p(u) has a noticeable effect on the sharpness of theswitching curve. This is illustrated in FIG. 10 where the peakdiffraction efficiency for θ_(ρ)=0 is plotted as a function of fieldusing different values of the standard deviation σ_(u). Although themaximum diffraction efficiency at E=0 is affected somewhat, the moredramatic effect appears in the sharpness of the switching. Therefore,observing the form of an experimental switching curve allows one to drawan inference about the statistics of the initial droplet-directororientation distribution. The diffraction efficiency approaches aminimum asymptotically for increasing field strength. The minimum isnear but not quite zero because the average droplet index approaches n₁,which is approximately equal to n_(p). If n±=n_(p), then the asymptotewould be zero. This switching by inducing a match of LC droplet index topolymer index is the classical type of switching observed in ordinaryPDLCs and is called index switching.

Alternatively, examining the same reflection grating at off-normalincidence, the plot in FIG. 11 shows peak diffraction efficiency as afunction of applied field for s-polarized, p-polarized, and unpolarizedlight. All of the parameters are the same as previously given, with theexception that the angle of incidence is θ_(p)=0.1π (18°. FIG. 11illustrates the difference in switching behavior between normal andoff-normal incidence. A true zero in diffraction efficiency is achievedfor p-polarization near E_(c), with efficiency then showing an increaseas the field is increased further. The diffraction efficiency fors-polarized light exhibits an asymptotic behavior similar to that seenat normal incidence. For unpolarized light, an average of the two curvesfor s and p polarization is displayed. It is difficult to obtain goodswitching behavior, i.e., high dynamic range, using unpolarized light atoff-normal incidence because of the disparity between s-polarization andp-polarization, consequently, in a preferred embodiment, p-polarizedlight may be used to yield the highest dynamic range.

The results given above are due to the tensor nature of the grating. Fors-polarization, the switching is asymptotic and based on index switchingas described above, where for large field values the droplet index is˜n_(⊥)≈n_(p). However, the case for p-polarization is quite differentand cannot be described as an index matching. Referring to Eq. (19), thecondition for the coupling coefficient κ_(p) to vanish is

$\begin{matrix}{{\tan\;\theta_{\rho}} = ( \frac{ɛ_{xx}^{(1)}}{ɛ_{zz}^{(1)}} )^{1/2}} & (23)\end{matrix}$and this is achieved at some particular value of E. The form of Eq. (23)is reminiscent of the definition of the Brewster angle for isotropicsystems and has an analogous physical interpretation. For light incidentfrom an isotropic medium of index n_(i) onto an isotropic medium ofindex n₂, the Brewster angle θ_(B) is the angle of incidence for whichthe reflectance of p-polarized light is zero. This can be calculatedfrom electromagnetic theory and is based on the conditions dictated byMaxwell's equations at the boundary between the two media, with tanθ_(B)=n₂/n_(i). At the Brewster angle, the rays transmitted to andreflected from the second medium are at a right angle. Brewster'scondition has thus been given the following interpretation. Forp-polarization, dipoles, i.e., oscillating electrons, are induced in thesecond medium in the plane of incidence and perpendicular to thetransmitted ray. These dipoles radiate and create a reflected ray backinto the first medium. However, dipoles do not radiate along a directionparallel to their direction of oscillation. At Brewster's angle, wherethe reflected and transmitted rays are at a right angle, the induceddipoles point along the direction of the reflected ray. Since they donot radiate any energy in this direction, the reflected ray vanishes.Although there is some controversy regarding this interpretation, it istrue that for non-conducting, non-magnetic media, the only work term inthe electromagnetic energy theorem that could contribute to thegeneration of an electric field E is the term proportional toRe(iωE·P*), where P is the dielectric polarization induced in themedium. If E and P are orthogonal, the work term is zero and no energycan be expended to generate the field E, even though the wave associatedwith E satisfies the boundary conditions. This certainly applies to thesituation of Brewster's law in isotropic media.

Applying a similar interpretation to Bragg diffraction in anisotropicmedia, in Eq. (19), the quantity ∈⁽¹⁾·ê_(ρ) is a vector pointing in thedirection of the spatially modulated part of the dielectric polarizationinduced by the reference field E_(ρ). If this vector is perpendicular toê_(σ), there can be no work done by the induced polarization to generatethe signal wave E_(σ), even though the direction of this wave isconsistent with the Bragg condition, and the coupling coefficientconsequently vanishes. This occurs in unslanted isotropic gratings at anincident angle of π/4 where the induced polarization, in this caseparallel to the reference field, is perpendicular to the signal field.For p-polarization the condition ê_(σ)·∈⁽¹⁾·ê_(ρ)=0 is equivalent to thecondition given by Eq. (23), which is induced by the applied field.Hence the applied field induces a dielectric polarization orthogonal tothe signal field to produce a zero in the diffraction efficiency, andthis is called polarization switching.

Polarization switching of p-polarized light can be put to use in makingan inverse mode HPDLC switchable reflection grating. An inverse modegrating is one for which the diffraction efficiency turns on to a highvalue when a voltage is applied. In normal mode HPDLC gratings a voltageturns the grating off (low diffraction efficiency). An inverse modegrating would be advantageous for certain applications, but is difficultto make using present materials. The concept is to orient thediffraction grating so that the internal incident angle of p-polarizedlight satisfies Eq. (23) at zero field. Thus the diffraction efficiencywould be zero. When a field is applied, ∈_(zz) ⁽¹⁾ increases while∈_(xx) ⁽¹⁾ decreases. By Eq. (19), the coupling coefficient increasesand the grating turns on. An example of the switching of s-polarized andp-polarized light for such a situation is illustrated in FIG. 12. Theconditions for this plot are: n_(p)=1.530, n_(⊥)=1.530, n_(∥)=1.750,α=0.5, f_(c)=0.9, L=15 μm, λ_(B)=0.525 μm, θ_(ρ)=0.188π (33.8°), ū=1,σ_(u)=0.4, q(φ₀)=0.5 (isotropic φ₀-distribution). The grating is turnedon for p-polarized light and off for s-polarized light at E—E_(c). Hencethe grating is in the inverse mode for p polarization and normal modefor s polarization.

A grating device as described herein can function as an electro-opticalpolarizing beam splitter. For example, at zero field incidentunpolarized light would be split into s-polarized light, i.e., reflectedand p-polarized light, i.e., transmitted. For an applied field>E_(c),the opposite effect would be achieved: unpolarized light would be splitinto s-polarized light that is now transmitted and p-polarized lightthat is now reflected.

In a first embodiment of the present invention, a system and method forcontrolling index modulation through nematic director control, isdescribed. For given LC birefringence and volume fraction, the indexmodulation can be maximized by maximizing the birefringence of the LCdroplets. This is achieved by distorting the droplets and aligning thesymmetry axes of each droplet in the same direction, which matches thepolarization direction of the incident light. It is possible to do thisby applying external stimuli that shape and orient the droplets as theyare formed in the phase separation process. Techniques for achievingthis using a magnetic field or an externally applied stress aredisclosed in U.S. Pat. No. 5,942,157 to Sutherland et al., which isincorporated herein by reference in its entirety.

This first embodiment describes a method for distorting the droplets andaligning the symmetry axes of each droplet in the same direction usingan electric field that is compatible with subsequent electricalswitching of the HPDLC optical device. The pre-polymer/LC material isplaced between glass plates with transparent electrodes as disclosed inU.S. Pat. No. 5,942,157. However, instead of transparent planarelectrodes, the electrodes are patterned as illustrated in FIG. 13 a.These are called interdigitated electrodes 30 or finger electrodes, withfinger height h 32 and finger separation b 34. These electrodeparameters, h and b, are adjusted for optimum performance. While theseparameters may vary between device applications, an exemplary dimensionis approximately 10 μm for both h and b, according to the relationshipthat the dimensions are approximately equivalent in size to thethickness of the HPDLC material. Both glass plates 36 a and 36 b areconfigured with interdigitated electrodes 30 a and 30 b, but the backplate 36 b electrodes 30 b are staggered with respect to the front plate36 a electrodes 30 a as illustrated in FIG. 13 b. In an alternativeembodiment, the back electrode could be a solid planar electrode (notshown). The pre-polymer LC material 38 is irradiated holographically asdisclosed in U.S. Pat. No. 5,942,157 to form either a reflection ortransmission hologram (not shown). However, while the system is beingcured, a voltage (V) 39 approximately equal to the switching voltage ofthe device is applied to every other finger electrode, with the samepattern applied to both front and back electrodes 30 a and 30 b. This isillustrated in FIG. 13 b along with the resulting electric fieldpattern. The fringing fields 40 of each electrode superpose in theholographic medium to create an in-plane electric field 42. This fieldorients the LC nematic directors 20 in the forming droplets 44 along thesame in-plane direction as the resulting electric field 42. This willalso slightly distort the droplets in this direction, making it theelastically favored direction at equilibrium. When the system reachesgelation and the voltages 39 are removed, this orientation is locked inplace. The resulting index modulation is maximized for incident lightpolarized in the same direction.

Alternatively, by applying different voltages, various degrees ofpolarization state between the parallel to the film plane to theperpendicular to the film plane may be achieved. The applied voltagesare determined by the switching voltage and are approximately equalthereto. For light polarized in the direction of the LC droplet symmetryaxes thus formed, the index modulation and hence the diffractionefficiency will be maximized. Light polarized perpendicular to thisdirection will have minimum diffraction efficiency.

To switch a hologram “off” that is recorded in this manner (i.e., tozero out or minimize the index modulation), the approximate switchingvoltage (V) 39 is now applied to each finger electrode in the front setof electrodes 30 a, with the back set of electrodes 30 b being connectedto ground. This produces the film-normal field pattern 46 illustrated inFIG. 14. The LC droplet symmetry axes of the switchable HPDLC material48 are thus reoriented in this direction, which produces the minimumindex modulation to incident light as illustrated in the figure. Thiselectrode configuration can also be used to optimize temporal responseas discussed further below.

In a second embodiment of the present invention, a system is describedfor controlling index modulation through fringe stability and/orcontrast control. Achieving excellent fringe stability and contrast inthe interferogram applied to the HPDLC material optimizes indexmodulation in holography. Fringe contrast is degraded by internalFresnel reflections in the cell containing the HPDLC material. Thesereflections also lead to the formation of cross gratings as the mainhologram is recorded, which contribute to haze and cosmetic defects anddecrease index modulation. The primary source of these reflections is atthe interface between the transparent electrode (i.e., indium tin oxide(ITO)) and the pre-polymer LC material. To alleviate this problem, abroad band anti-reflection (AR) coating is incorporated into thetransparent electrodes. The term AR coating refers to a substantiallytransparent multilayer film that is applied to optical systems (e.g.,surfaces thereof) to substantially eliminate reflection over arelatively wide portion of the visible spectrum, and thereby increasethe transmission of light and reduce surface reflectance. Knownanti-reflection coatings include multilayer films comprising alternatinghigh and low refractive index materials (e.g., metal oxides) asdescribed, for instance, in U.S. Pat. Nos. 3,432,225, 3,565,509,4,022,947, and 5,332,618 which are incorporated herein by reference intheir entireties. In cases of the present invention where etching of thetransparent electrode is not necessary, an AR coating is obtained by athin film stack of alternating layers of magnesium fluoride and ITOapplied to the glass on the side facing the pre-polymer LC material. Forcases where etching of the ITO is desired, the preferred AR coating is athin film stack of tantalum oxide/magnesium fluoride deposited on theglass with an ITO overcoat. In addition, during the recording process,an AR-coated piece of glass is optically connected to the outside facesof the holographic cell using index-matching fluid. These AR coatingsare optimized for the wavelength used in recording the hologram. Oneskilled in the art recognizes the variations in AR coating that may bealternatively used in this embodiment of the present invention.

In a third embodiment of the present invention, haze and cosmeticquality, and diffraction efficiency are controlled throughpre-establishment of a loosely gelled network in the PDLC recordingmedium. As mentioned earlier, the rapid polymerization and elasticrelaxation of the multi-functional acrylate system can lead toinstability and non-uniformity in the optical quality of the hologram,as illustrated by the example in FIG. 15 for a transmission hologram.The two-lobed 49 “walnut-shaped” pattern observed in FIG. 15 is a resultof the instability described above. This is related to the rapidformation of a gel network in this free-radical system and non-uniformshrinkage of the polymer set in place by the formation of the hologram.In reflection holograms, this non-uniform shrinkage leads to non-uniformchirp and tapering of the index modulation, producing a broadening ofthe diffraction notch, a reduction of the peak diffraction efficiency,and a washing out of the sidelobes. Consequently, in this thirdembodiment of the present invention, a pre-establishment of a looselygelled network prior to hologram recording is accomplished in a varietyof ways. This loosely gelled network is (a) not so stiff that itinhibits the diffusion of components and subsequent phase separationwhich are crucial to the formation of a switchable H-PDLC hologram, but(b) is sufficiently strong to stabilize the system and prevent shrinkageinstabilities from setting in as the hologram begins to form. One way toaccomplish this, for example, is by blocking one of the two beamsutilized in the recording setup for a period of approximately 2-5seconds so that the first exposure of the sample is a beam of uniformamplitude and phase that irradiates the sample uniformly, such that thisradiation partially bleaches the photoinitiator dye uniformly throughoutthe sample. This partial bleaching can also be done by blocking bothcoherent beams and irradiating the sample with an incoherent beam ofradiation. This partial bleaching of the photoinitiator establishes aloose gel network. After this process, both coherent beams are unblockedso that the sample is irradiated holographically in the usual manner.The hologram is then recorded in an identical manner as previouslydescribed in, for example, U.S. Pat. No. 5,942,157. The result is aswitchable hologram of high diffraction efficiency and excellent opticalas well as cosmetic quality, with uniform diffraction efficiency acrossthe sample.

As an alternative to the photoinitiator partial bleaching techniqueabove, wherein the photoinitiator matched to the recording wavelength ispartially bleached, additional photoinitiators can be added to thepre-polymer material so that pre-establishment of the loosely gellednetwork can be accomplished using illumination by a wavelength that doesnot overlap with the absorption spectrum of the photoinitiator matchedto the laser recording wavelength. Examples may include usingultraviolet (“UV”) initiators to expose the PDLC recording medium forshort periods of time with UV illumination, or using visible initiatorsthat do not interfere with hologram recording. A specific exampleincludes adding methylene blue to a sample to be recorded with 488-nmlight from an argon-ion laser. This sample is exposed 632.8 nm lightfrom a He—Ne (helium neon) laser prior to holographic recording withoutbleaching the initiator that is sensitive to the 488-nm radiation.

It is understood that the pre-establishment of a loosely gelled networkis not limited by the radiation exposure methods described above. Anytechnique to gently and partially cure the sample so that a loose gelnetwork is established is contemplated by this disclosure. Thesetechniques are known to those in the art of polymer chemistry and mayinclude heat, electron beams, or the presence of other reactants thatcan be triggered by some external mechanism. The third embodimentdescribes the formation of a loosely gelled network prior to hologramrecording in order to stabilize the system against non-uniform shrinkageas the hologram forms during a subsequent photopolymer chemicalreaction.

Another alternative method commensurate with the scope of the thirdembodiment comprises loading the pre-polymer PDLC recording medium intoa pre-existing loose network, such as an aerogel. An aerogel is a glassor polymer network that consists mostly of air voids that are muchlarger than a typical grating period or LC droplet in an H-PDLC. Thepre-polymer PDLC recording medium fills the voids by capillary action.Such a filled aerogel is then sandwiched between two ITO-coated glassplates and irradiated holographically in the manner describedpreviously. The aerogel does not prevent diffusion of components orsubsequent phase separation of LC droplets in the grating planes, butacts analogously to the loosely gelled polymer network of the previousexamples to stabilize the system against non-uniform shrinkage.

This technique of pre-establishment of a loosely gelled network in thesample can be applied to transmission and reflection gratings alike.This technique decreases the haze, improving the optical quality ofholograms. Further, this technique stabilizes the system to shrinkagenormal to the plane of the film, which reduces chirping of the gratingperiod and tapering of the index modulation profile. This enhances thediffraction efficiency of the hologram. An example of improvement of thediffraction efficiency in a reflection hologram using this technique bypartial bleaching of the photoinitiator is given in FIG. 16.

In a fourth embodiment of the present invention, haze and cosmeticquality are controlled using index matching and scattering control.Significant cosmetic inhomogeneity and haze can be attributed to thepresence of cross-gratings that appear as a result of reflections fromthe internal and external surfaces of the HPDLC optical device duringrecording of the hologram. These reflections interfere with both theincident beams and other reflections, thus recording unwanted hologramsin the HPDLC film. In order to minimize these unwanted reflections,conductive index-matched transparent electrodes as described withreference to the third embodiment are utilized. This greatly reducesunwanted internal reflections. These anti-reflective electrodes reducereflection from the internal surfaces. Second, a transparent tankrecording setup is employed to greatly reduce reflections from theexternal surfaces.

Unwanted reflections at the glass/air interfaces are rendered harmlessby the transparent tank arrangement depicted in FIG. 17. It is widelyknown in the holographic industry that these reflections aretroublesome, thus many organizations record holograms in tanks ofindex-matching fluid. While this approach can be effective, it islabor-intensive and requires extensive clean up. In addition, with PDLCmaterials, index-matching fluid can dissolve the LC, and therefore thecells must be completely sealed if such an approach is to be used. Thetransparent tank arrangement 50 depicted in FIG. 17 uses prisms 52 orglass blocks and neutral density (ND) filters 54 to stop unwantedreflections from exposing the holographic cell 56. In a specificembodiment, two custom BK-7 blocks 52 possessing the same refractiveindex as the HPDLC optical device 56 are manufactured to provide aparticular holographic geometry known to those skilled in the art. AHPDLC optical device 56 is placed in optical contact between the twoblocks, usually with a drop or two of index-matching fluid. If aswitchable HPDLC hologram is to be recorded, anti-reflective transparentelectrodes are used.

As with a bare cell, the reflection at either first glass/air interface58 is reflected safely away. In FIG. 17 the first glass/air interfaces58 are angled at 13° and 19°, respectively. The reflection mostproblematic is the second surface reflection which, in a bare cell,travels back through the film. ND filters 54 are placed in opticalcontact at the second glass/air interfaces, separated by index matchingfluid 60, opposite the 13° and 19° angled faces, where the recordingbeams 62 exit. Here, the ND filters 54 safely absorb the recording laserbeams 62 before a significant reflection occurs. One skilled on the artrecognizes the various optical densities that are available for use asND filters (e.g., 3 OD). With this arrangement, only a few drops ofindex matching fluid are needed, less if the ND filters are bonded tothe block. Thus, this arrangement represents an improvement over the useof an entire tank of index-matching fluid, especially considering thevulnerability of the HPDLC optical device to these fluids. Utilizing thetransparent tank arrangement 50, baseline transmission of HPDLCholograms can be increased by a significant percentage, e.g., as much as10-15%. This means less haze, less backscatter, and a cosmeticallyimproved HPDLC optical device. The negation of unwanted secondarygratings leads to an improved diffraction efficiency.

According to a fifth embodiment of the present invention, switchingvoltage can be controlled via tailoring of LC droplet size and shape. Byway of background, switching is best discussed in the context of asimple model. According to U.S. Pat. No. 5,942,157, the switchingvoltage of a switchable hologram is related to the critical electricfield (E_(r)) necessary to reorient the LCs. This critical field isgiven by the following equation:

$\begin{matrix}{E_{c} = {\frac{1}{3a}{( {\frac{\sigma_{LC}}{\sigma_{P}} + 2} )\lbrack \frac{\overset{\_}{k}( {l^{2} - 1} )}{\Delta ɛ} \rbrack}^{1/2}}} & (24)\end{matrix}$Equation (24) predicts the critical field for an elongated LC droplet,with semi-major axis a, semi-minor axis b, and aspect ratio l=a/b.Further to equation (1), σ_(LC) and σ_(p) are the electricalconductivities of the LC and polymer, respectively; k is an averageelastic force constant while Δ∈ is the dielectric anisotropy, bothconsidered constant properties of the bulk LC. This equation can be usedto identify properties to target for reducing the switching voltage. Theaspect ratio l can be controlled, but may be traded off against otherparameters, e.g., polarization dependence or index modulation.The same elongated droplet model leading to Equation (24) predicts arelaxation time, when the applied field is turned off, given by

$\begin{matrix}{\tau_{off} = \frac{\gamma_{1}a^{2}}{\overset{\_}{k}( {l^{2} - 1} )}} & (25)\end{matrix}$where γ₁ is the rotational viscosity coefficient of the LC. Thus, areduction in the effective elastic force constant that produces areduction in the critical field by a factor of M will tend to increasethe relaxation time by a factor of M². If the longer relaxation time isstill compatible with the switching time needed for a particularapplication, then the slower relaxation is not a severe penalty.However, there may be cases where a longer relaxation time is notdesired.

The limiting speed of the switchable hologram is given by the relaxationtime τ_(off) given by Equation (25) above. Two important geometricalparameters are droplet size a and shape l. Droplet size a also impactsscattering loss; the scattering coefficient increases with sizeapproximately as a³. There is also a trade-off of switching voltage withrelaxation time as seen in Equation (24). It is clearly desirable tokeep a as small as possible. Since scattering and relaxation time areapproximately proportional to a³ and a², respectively, while switchingvoltage is proportional to a⁻¹, much is to be gained by minimizing a.Ultimately, though, this will begin to increase switching voltageunfavorably, even when optimizing matrix conductivity, interfacialanchoring, and effective dielectric anisotropy. At some point it isdesirable to offset decreases in a with some other parameter.

One such off-set parameter is the droplet shape. Changes in size (Δa/a)can be approximately offset by corresponding changes in shape (Δl/l), ascan be seen by reference to Equations (24) and (25). Distorting dropletswhile they are being formed during phase separation can control dropletshape. It is generally desirable to induce distortion (i.e., elongation)in a direction parallel to the holographic film plane. Techniques forachieving this using external magnetic fields and stress fields havebeen discussed in U.S. Pat. No. 5,942,157. The first embodiment of thepresent invention sets forth a technique for controlling LC dropletformation using interdigitated electrodes and in-plane electric fields.In this embodiment, a is minimized to reduce scatter and relaxationtime. Alternatively, if it is necessary to then increase a to optimizeswitching voltage, then e can be increased simultaneously to prevent therelaxation time from increasing. This off-setting procedure allows forLC droplet formation that optimizes HPDLC optical device operation.

According to a sixth embodiment of the present invention, switchingspeed is controllable through electrode design and voltage drive scheme.The limiting speed of the switchable hologram is given by the relaxationtime τ_(off) given by Equation (25) above. The response time (i.e., whenvoltage is applied) is field dependent, however. Under conditions ofoptimal switching where (2+σ_(LC)/σ_(p))/3˜1, the response time when thecritical field is applied can be estimated from

$\begin{matrix}{{ \tau_{on} \sim\frac{\gamma_{1}}{4{\Delta ɛ}\; E_{c}^{2}}}.} & (26)\end{matrix}$However, for large fields E as compared to E_(c) (E>>E_(c)) the responsetime is approximately given. by

$\begin{matrix}{{ \tau_{on} \sim\frac{\gamma_{1}}{{\Delta ɛ}\; E^{2}}}.} & (27)\end{matrix}$Therefore, by way of example, assuming γ₁=0.27 kg/m-s and Δ∈=15.36∈₀, aresponse time of 10 μs would require a field strength of ˜15 V/μm. Thisanalysis indicates that a fast response time would be achievable if thehologram could be driven both “on” and “off” with a large enough field.Pursuant to the sixth embodiment of the present invention, this can beachieved while maintaining low power consumption.

Referring to FIG. 18 a, to drive the hologram “off,” a fieldperpendicular to the film plane is applied. This is done by applying avoltage (V) 39 approximately equal to the switching voltage to eachfinger electrode 30 a on the front plate 36 a, and connecting theelectrodes 30 b on the back plate 36 b to ground. This is similar to theeffect that occurs when the front and back electrodes are planar ratherthan patterned, and the field that results is illustrated through fieldlines 40. To drive the hologram back “on,” the previous voltage schemeis removed, and simultaneously a new voltage scheme is applied, asillustrated in FIG. 18 b. Referring to FIG. 18 b, the voltage (V) 39 onevery other finger electrode 30 a on both plates 36 a, 36 b isapproximately equal to the switching voltage, with intermediate fingerelectrodes 30 b on each plate 36 a, 36 b being connected to ground. Thisproduces an in-plane electric field 42 and drives the hologram “on.”Switching back and forth between these two schemes drives the hologram“on” and “off,” with a response time in each case given approximately byEquation (27).

Switchable HPDLC holograms are normally driven at 500-2000 Hz. Theperiod of this waveform (0.5-2 ms) is long compared to the desiredresponse time of the device. Therefore, the hologram can be overdrivenin the first cycle of the waveform by a field sufficient to produce afast response time given by Equation (27), with the rest of the waveformsettling to a lower value ˜E_(c) to maintain the desired state of thehologram. This type of waveform is illustrated in FIG. 19. In thismanner, the voltage is retained at a reasonably low value during most ofthe operation of the device, with little increase in the powerconsumption.

The embodiments described above are not intended to be limiting. Oneskilled in the art recognizes the obvious variations and trade-offs thatare included within the scope of the embodiments set forth herein.

1. A method for controlling an index modulation of a switchable polymerdispersed liquid crystal optical component comprising: providing apre-polymer liquid crystal material between a first and secondsubstrate, the first and second substrate having a first and secondelectrode pattern thereon, respectively, for applying a switchingvoltage to the switchable polymer dispersed liquid crystal opticalcomponent, wherein at least one each of the first and second electrodepatterns consists of interdigitated electrodes; applying a voltageapproximately equal to the switching voltage to only every otherinterdigitated electrode, creating an in-plane electric field within thepre-polymer liquid crystal material; holographically irradiating thepre-polymer liquid crystal material resulting in polymerization of thepre-polymer liquid crystal, wherein liquid crystal droplets formed fromthe holographic irradiation are formed with symmetry axes oriented in asame direction as the in-plane electric field; and removing the voltageapproximately equal to the switching voltage once polymerization iscomplete.
 2. The method according to claim 1, wherein the interdigitatedelectrodes of the first electrode pattern are staggered with respect tothe interdigitated electrodes of the second electrode pattern.
 3. Themethod according to claim 1, wherein at least one of the first andsecond electrode patterns is a solid planar pattern.
 4. The methodaccording to claim 1, wherein each of the interdigitated electrodes hasa height (h) and each of the interdigitated electrodes are separated bya separation (b), wherein each of (h) and (b) are approximately equal tothe width of the polymer dispersed liquid crystal optical material. 5.The method according to claim 4, wherein (h) is approximately 10 μm. 6.The method according to claim 4, wherein (b) is approximately 10 μm. 7.The method according to claim 1, wherein the first electrode pattern istransparent.
 8. The method according to claim 1, wherein a firstfringing field of a first electrode superposes a second fringing fieldof a second electrode to create an in-plane electric field.
 9. Themethod according to claim 8, wherein the in-plane electric field orientsa plurality of nematic directors in a plurality of forming dropletsalong the same in-plane direction as the electric field.
 10. The methodaccording to claim 9, wherein the plurality of droplets are distorted inthe direction of the electric field at equilibrium.
 11. The methodaccording to claim 10, wherein the orientation of the nematic directorsis established upon gelation and removal of the applied voltage.